System and Method for Medical Predictive Models Using Likelihood Gamble Pricing

1. A method for predicting survival rates of medical patients, said method comprising the steps of: providing a set D of survival data for a plurality of medical patients having a same condition; providing a regression model, said model having an associated parameter vector β; providing an example x 0 of a medical patient whose survival probability is to be classified; calculating a parameter vector {circumflex over (β)} that maximizes a log-likelihood function of β over the set of survival data, l(β|D), wherein the log likelihood l(β|D) is a strictly concave function of β and is a function of the scalar xβ; calculating a weight w 0 for example x 0 ; calculating an updated parameter vector β* defined as the parameter vector β that maximizes a function l(β|D∪{(y 0 ,x 0 ,w 0 )}), wherein data points (y 0 ,x 0 ,w 0 ) augment said set D; calculating a fair log likelihood ratio λ f from {circumflex over (β)} and β* using λ f =λ(β*|x 0 )+sign(λ({circumflex over (β)}|x 0 )){l({circumflex over (β)}|D)−l(β*|D)}; and mapping the fair log likelihood ratio λ f to a fair price y 0 f , wherein said fair price is a probability that class label y 0 for example x 0 has a value of 1.

2. The method of claim 1 , wherein said weight w 0 is calculated from w ◇ = - ∂ ∂ ( x ◇ ⁢ β ) ⁢  λ ⁡ ( β | x ◇ )  ∂ ∂ ( x ◇ ⁢ β ) ⁢ l ⁡ ( β | ( y ◇ , x ◇ , w ◇ = 1 ) ) ⁢ | β = β ^ , wherein λ(β|x 0 ) is a log-likelihood ratio of a likelihood that a class label y 0 as a value of 1 over a likelihood that a class label y 0 has a value of 0, wherein said log-likelihood-ratio is an affine function of the scalar xβ.

3. The method of claim 2 , wherein said regression model is a logistic regression model with a probability of label y being 1 is p ⁡ ( y = 1 | x , β ) = 1 1 + exp ⁡ ( - λ ⁡ ( β | x ) ) , ⁢ wherein ⁢ ⁢ λ ⁡ ( β / x ) = x ⁢ ⁢ β .

4. The method of claim 3 , wherein said log-likelihood of β is l ⁡ ( β | D ) = ∑ i = 1 N ⁢ w i ⁢ { ( y i - 1 ) ⁢ x i ⁢ β - log ⁡ ( 1 + exp ⁡ ( - x i ⁢ β ) ) } .

5. The method of claim 3 , wherein said weight w 0 is calculated from w ◇ = - sign ⁡ ( x ◇ ⁢ β ^ ) y ◇ - 1 + 1 1 + exp ⁡ ( x ◇ ⁢ β ^ ) .

6. The method of claim 3 , wherein said fair log likelihood ratio λ f is λ f =x 0 β*sign(x 0 {circumflex over (β)})└l({circumflex over (β)}|D)−l(β*|D)┘.

7. The method of claim 3 , wherein said fair price is y ◇ f = 1 1 + exp ⁡ ( - λ f ) .

8. The method of claim 2 , wherein said regression model is a Gaussian regression model with two clusters having a Gaussian distribution for either class, N(0,σ 2 ) and N(1,σ 2 ), wherein σ 2 is a standard deviation.

9. The method of claim 8 , wherein said log-likelihood of β is l ⁡ ( β | D ) = - ∑ i = 1 N ⁢ w i ⁡ ( x i ⁢ β - y i ) 2 / σ 2 .

10. The method of claim 8 , wherein said weight w 0 is calculated from w ◇ = - sign ⁡ ( 2 ⁢ x ◇ ⁢ β ^ - 1 ) x ◇ ⁢ β ^ - y ◇ .

11. The method of claim 8 , wherein said fair log likelihood ratio λ f is λ f = log ⁢ ⁢ p ⁡ ( y = 1 | x , β ) p ⁡ ( y = 0 | x , β ) = ( 2 ⁢ x ⁢ ⁢ β - 1 ) / σ 2 .

12. The method of claim 8 , wherein said fair price is y ♦ f = ⁢ 1 + λ f ⁢ σ 2 / 2 = ⁢ x ♦ ⁢ β * + sign ⁡ ( 2 ⁢ x ♦ ⁢ β ^ - 1 ) 2 ⁢ ( l ⁡ ( β ^ | D ) - l ⁡ ( β * | D ) ) .

13. The method of claim 1 , wherein said weight w 0 =2, and said updated parameter vector β* is determined by maximizing l(β|D∪{(1,x 0 ,1),(0,x 0 ,1)}), wherein (1,x 0 ,1),(0,x 0 ,1) are data points augmenting said set D.

14. A method for predicting survival rates of medical patients, said method comprising the steps of: providing a set D of survival data for a plurality of medical patients having a same condition; providing a regression model, said model having an associated parameter vector β; providing an example x 0 of a medical patient whose survival probability is to be classified; calculating a first parameter l 1 = max β ⁢ l ⁡ ( β | D ⋃ { ( 1 , x ♦ , 1 ) } ) that maximizes a log-likelihood function of β over the set of survival data, l(β|D) augmented by a data point (1,x 0 ,1), wherein the log likelihood l(β|D) is a strictly concave function of β and is a function of the scalar xβ; calculating an second parameter l 0 = max β ⁢ l ⁡ ( β | D ⋃ { ( 0 , x ♦ , 1 ) } ) that maximizes a log-likelihood function of β over the set of survival data, l(β|D) augmented by a data point (0,x 0 ,1); calculating a fair log likelihood ratio λ f from λ f =l 1 −l 0 ; and mapping the fair log likelihood ratio λ f to a fair price y 0 f , wherein said fair price is a probability that class label y 0 for example x 0 has a value of 1.

15. A non-transitory program storage device readable by a computer, tangibly embodying a program of instructions executable by the computer to perform the method steps for predicting survival rates of medical patients, said method comprising the steps of: providing a set D of survival data for a plurality of medical patients having a same condition; providing a regression model, said model having an associated parameter vector fi; providing an example x:, of a medical patient whose survival probability is to be classified; calculating a parameter vector/) that maxirnizes a log-likelihood function of fi over the set of survival data, I(/?ID), wherein the log likelihood t(./]ID) is a strictly concave function of fl and is a function of the scalar aft; calculating a weight w: for example x ; calculating an updated parameter vector fi* defined as the parameter vector fi that maximizes a function t(fll D t.9{(3′,:r˜, ˜1;:)}), wherein data points (y: ,x˜, ˜.;) augment said set D; calculating a fair log likelihood ratio 2 from /˜ and fi* using 2c:=d.(fl* [ a′.)+sign(/˜(fl [ a.˜)˜l(/] I D)−l(fl* t D)}; and mapping the fair log likelihood ratio 2f to a fair price Y;!l, wherein said fair price is a probability that class label y+ for example x:> has a value of 1.

16. The computer readable program storage device of claim 15 , wherein said weight w 0 is calculated from w ♦ = - ∂ ∂ ( x ♦ ⁢ β ) ⁢  λ ⁡ ( β | x ♦ )  ∂ ∂ ( x ♦ ⁢ β ) ⁢ l ⁡ ( β | ( y ♦ , x ♦ , w ♦ = 1 ) ) ⁢ | β = β ^ , wherein λ(β|x 0 ) is a log-likelihood ratio of a likelihood that a class label y 0 has a value of 1 over a likelihood that a class label y 0 has a value of 0, wherein said log-likelihood-ratio is an affine function of the scalar xβ.

17. The computer readable program storage device of claim 16 , wherein said regression model is a logistic regression model with a probability of label y being 1 is p ⁡ ( y = 1 | x , β ) = 1 1 + exp ⁡ ( - λ ⁡ ( β | x ) ) , wherein ⁢ ⁢ λ ⁡ ( β | x ) = x ⁢ ⁢ β .

18. The computer readable program storage device of claim 17 , wherein said log-likelihood of β is l ⁡ ( β | D ) = ∑ i = 1 N ⁢ w i ⁢ { ( y i - 1 ) ⁢ x i ⁢ β - log ⁡ ( 1 + exp ⁡ ( - x i ⁢ β ) ) } .

19. The computer readable program storage device of claim 17 , wherein said weight w 0 is calculated from w ♦ = - sign ⁡ ( x ♦ ⁢ β ^ ) y ♦ - 1 + 1 1 + exp ⁡ ( x ♦ ⁢ β ^ ) .

20. The computer readable program storage device of claim 17 , wherein said fair log likelihood ratio λ f is λ f =x 0 β*+sign(x 0 {circumflex over (β)})└l({circumflex over (β)}|D)−l(β*|D)┘.

21. The computer readable program storage device of claim 17 , wherein said fair price is y ♦ f = 1 1 + exp ⁡ ( - λ f ) .

22. The computer readable program storage device of claim 16 , wherein said regression model is a Gaussian regression model with two clusters having a Gaussian distribution for either class, N(0,σ 2 ) and N(1,σ 2 ), wherein σ 2 is a standard deviation.

23. The computer readable program storage device of claim 22 , wherein said log-likelihood of β is l ⁡ ( β | D ) = - ∑ i = 1 N ⁢ w i ⁡ ( x i ⁢ β - y i ) 2 / σ 2 .

24. The computer readable program storage device of claim 22 , wherein said weight w 0 is calculated from w ♦ = - sign ⁡ ( 2 ⁢ x ♦ ⁢ β ^ - 1 ) x ♦ ⁢ β ^ - y ♦ .

25. The computer readable program storage device of claim 22 , wherein said fair log likelihood ratio λ f is λ f = log ⁢ ⁢ p ⁡ ( y = 1 | x , β ) p ⁡ ( y = 0 | x , β ) = ( 2 ⁢ x ⁢ ⁢ β - 1 ) / σ 2 .

26. The computer readable program storage device of claim 23 , wherein said fair price is y ♦ f = ⁢ 1 + λ f ⁢ σ 2 / 2 = ⁢ x ♦ ⁢ β * + sign ⁡ ( 2 ⁢ x ♦ ⁢ β ^ - 1 ) 2 ⁢ ( l ⁡ ( β ^ | D ) = l ⁡ ( β * | D ) ) .

27. The computer readable program storage device of claim 15 , wherein said weight w 0 =2, and said updated parameter vector β* is determined by maximizing l(β|D∪{(1,x 0 ,1),(0,x 0 ,1)}), wherein (1,x 0 ,1),(0,x 0 ,1) are data points augmenting said set D.